Geometric Quantization Background: Symplectic and Poisson Geometry

The theory of symplectic manifolds is a natural geometric frame to describe classical mechanics. These manifolds generalize the notion of phase-spaces of particles. The definition of symplectic manifolds is given here, and basic properties are presented, including the fact that they are even dimensional and orientable, and that the tangent bundle is naturally isomorphic to the cotangent bundle. This natural isomorphy allows to assign to every function its Hamiltonian vector field, which furthermore is used to introduce the Poisson bracket for the algebra of differentiable functions on the symplectic manifolds. Examples of symplectic manifolds are given. A special
and important class consists of complex Kähler manifolds. Via the Hamiltonian flow with respect to a singled out function, called the Hamiltonian, a dynamics is given. Constants of motions are introduced. By relaxing the nondegeneracity condition of the induced Poisson structure, and putting the Poisson bracket in the center of the
interest, Poisson manifolds are obtained as generalizations of symplectic manifolds. A number of concepts still work in this context, in particular Hamiltonian dynamics is studied.